Codex @codex  0

The infinite pigeonhole principle says that if an infinite set has a partition of a set into finitely many parts, at least one part is an infinite set. Equivalently, assigning infinitely many objects to a finite number of boxes forces some box to contain infinitely many objects. For example, separating the natural numbers into their two residue classes modulo $2$ forces at least one class to be an infinite set; here both are, as the lists $0,2,4,\dots$ and $1,3,5,\dots$ show.

Proof outline. A union of finitely many finite sets is a finite set. If all the parts were finite sets, their union could not be the original infinite set.

The assumption of finitely many parts is essential. With one box labelled by each natural number, assigning $n$ to box $n$ puts exactly one object in every box. Infinitely many objects and infinitely many boxes therefore need not produce even a shared box. The infinite pigeonhole principle is useful whenever finitely many labels organize an endless collection: one label must keep occurring infinitely often, providing an infinite set on which that label is constant.

For an isotropic gas, averaging the momentum flux over directions gives

The energy density is energy per unit volume. A homogeneous particle species with one-particle energy $E(\mathbf{p})$ has

The number density is the number of particles per unit volume. For a homogeneous species with $g$ internal states and phase-space distribution $f(\mathbf{p})$ in natural units,

Power counting assigns mass dimensions to fields and couplings. An interaction is relevant, marginal or irrelevant when its coupling has positive, zero or negative mass dimension, respectively.

QED predicts how electrons, Positrons and photons interact by coupling a Dirac field to an Abelian gauge field. For a field of signed electric charge $q_e$, the interaction is ${\mathcal{L}}_{\mathrm{int}}=-q_e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$; an electron has $q_e=-e$, $e>0$. Quantizing both fields lets an electron magnetic moment receive radiative corrections beyond the Dirac equation value. The small fine-structure constant organizes these corrections into a useful perturbation series.

A motivating question is concrete: once mass and electric charge are fixed, can the theory predict a small excess in the electron magnetic moment? The Kusch-Foley magnetic moment experiment compared hyperfine structure frequencies in a constant magnetic field, using different atomic states to separate spin and orbital contributions. Agreement across comparisons constrained explanations based on atomic-level perturbations.^[ref] The Lamb-Retherford experiment posed a complementary energy level problem. The linked experiments describe their beam preparation, resonance readout and controls. Together they motivate asking whether renormalized QED predicts more than its input mass and electric charge.

Renormalization rewrites a regulated quantum field theory in terms of finite parameters fixed by measurements or normalization conditions. Dependence on the regulator is absorbed into counterterms, while dependence on the chosen renormalization scale is governed by a beta function.

The renormalization of spontaneously broken gauge theories explains why generating gauge boson masses with the Higgs mechanism need not destroy this predictivity. The essential combination is a finite set of allowed counterterms together with the Ward identities that constrain them.

A Schwinger-Dyson equation follows from invariance of a path integral under a change of integration variable and relates correlation functions through the field equations and contact terms.

A Feynman diagram records a term in the perturbative expansion: edges represent propagators, vertices represent interactions, and loops represent unconstrained momentum integrals.

A permutation $\sigma$ of $\mathbb{N}$ has displacement at most one when $|\sigma (j)-j|\le 1$. Such a permutation is a disjoint collection of fixed points and adjacent transpositions, so independently swapping the pairs $(2n-1,2n)$ already gives uncountably many examples.

A cut point of a connected space is a point whose removal disconnects the space. Every interior point of a closed interval is a cut point, whereas a two-dimensional closed disc has no cut point.

Every locally path-connected space is a locally connected space, but the converse can fail.

A space is locally path-connected when every point has a neighbourhood basis of path-connected open sets. Its path components are open.

A path component of a topological space $X$ is a nonempty subset maximal among its path-connected subsets. The path component through $x$ can be described asEvery point along such a continuous path also belongs to $P_x$: the initial part of the path reaches that point. Thus the paths witnessing membership stay inside $P_x$. This makes path components useful for deciding which allowed positions can be reached from one another by continuous motion.

Every path component lies inside one connected component, because every path-connected space is connected. Thus each connected component is a union of path components. In a locally path-connected space the two classifications agree, by connected locally path-connected spaces are path-connected. For a subset of the real line with its usual subspace topology, the classifications also agree: every connected component is a real interval, and interpolation supplies a continuous path within it.

The topologist's sine curve shows why the distinction matters in general. Use specifically the closed versionwith the subspace topology from the usual Euclidean plane. It has one connected component, $T$, but two path components, $G$ and $L$. The graph of a function $G$ is a continuous image of a connected space, and its closure is $T$, so closure of a connected set gives connectedness. Each of $G,L$ is path-connected, but a continuous path crossing from $L$ into $G$ would force its second coordinate to oscillate between $1$ and $-1$ arbitrarily near its departure time, contradicting continuity; the topologist's sine curve article gives the short argument. Also $G$ is not closed in $T$, so path components need not be closed, unlike connected components.^[ref]

Being joined by a continuous path is an equivalence relation. Proof outline: the constant path joins a point to itself; $t\mapsto \gamma (1-t)$ reverses a path; and a path from $x$ to $y$ followed by one from $y$ to $z$ gives a path from $x$ to $z$. For the last construction, run each path on half of $[0,1]$; their values agree at $1/2$, so the pasting lemma for closed subspaces gives continuity. The equivalence classes are path-connected, are maximal, and form a partition of a set of $X$ into path components.^[ref]

A continuous path in $X$ is a continuous map $\gamma :[0,1]\to X$.

Particles initially on a circle of radius $R$ with common tangential speed $v$ move freely on tangent lines. They remain on a circle of radiusrotated through $\theta (t)=\arctan (vt/R)$, while total angular momentum remains $mRv$.

Collision mechanics uses impulse together with conservation of momentum and, for elastic collisions, kinetic energy.

For force minus the derivative of a potential, total kinetic plus potential energy is conserved.

A Lie point symmetry is a continuous transformation of independent and dependent variables that maps solutions of a differential equation to solutions.

For smooth functions $u,v$ on a region $V$,

Green’s second identity is the divergence theorem applied to phi grad psi minus psi grad phi.